We will need analogous results for symmetric tensor fields. For a simply connected open set \(\Omega \) , given a smooth symmetric tensor field \(a \in C^\infty (\Omega ,\operatorname{Sym})\) , there exists \(u\in C^\infty (\Omega ,\mathbb {R}^2)\)  satisfying \(a = (1/2) \left(\nabla u+(\nabla u)^T\right)\)  if and only if \(\operatorname{curl}\operatorname{curl}a=0\) . More generally, given \(A \in \mathcal {D}^{\prime }(\Omega ,\operatorname{Sym})\) , there exists \(U\in \mathcal {D}^{\prime }(\Omega ,\mathbb {R}^2)\)  satisfying \(A = (1/2) \left(\nabla U+(\nabla U)^T\right)\)  if and only if \(\operatorname{Curl}\operatorname{Curl}A=0\)  [1]}. The following lemma establishes a stronger result for the existence and regularity of a distributional vector field given a piecewise continuous (distributional) symmetric tensor field.
